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Effect of white noise on the \(d\)-dimensional harmonic oscillator. (Effet d’un bruit blanc sur l’oscillateur harmonique de dimension \(d\).) (French) Zbl 0904.60039
The author studies the \(d\)-dimensional isotropic harmonic oscillator perturbed by small in some sense random term \(dW_tx_t\), where \(W_t\) denotes the Brownian motion. He obtains explicit formulas for the limit of Lyapunov type exponents \(t^{-1}\ln E(\| x_t\|^2)\) and almost sure limit of \(t^{-1}\ln(\| x_t\|^2+\| v_t\|^2)\), when the small parameter tends to zero.
MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
93E15 Stochastic stability in control theory
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